single word or phrase when one of the department manager tries "blur the lines of responsibility"
Solution 1:
Such activity may be termed stepping/treading on someone's toes.
If you step or tread on someone's toes, you say or do something that upsets or annoys that person, especially by involving yourself in matters that are not your responsibility.
[Cambridge Dictionary]
From Facebook.Global ELT (re-ordered]:
To tread on someone's toes is to offend or insult someone, especially by trespassing on [their] field of responsibility.
- His colleagues felt that he was treading on their toes.